The structure of \(\Pi\) spaces. I
Let \(\Pi\) denote a Krein space with an inner product \((.,.)\). A decomposition \(\Pi =H_+\oplus H_-\) is called a regular decomposition if \(H_{\pm}\) are Hilbert spaces relative to the inner products \((.,.)\). Let \(A\) denote a linear operator from the Hilbert space \(H_-\) to \(H_+\). Let \(L_ A=\{x+Ax,\) \(x\in D(A)\equiv\) domain of \(A\)\}. The author here investigates various properties of \(A\). He then uses these ideas to study the direct sum decomposition of \(\Pi\)-spaces and corresponding projections. In particular, the author gives a new proof of the following: Theorem: A linear subspace \(L\) of \(\Pi\) is a complete subspace (i.e. a \(\Pi\)-type space and closed) iff \(\pi =L\oplus L^{\perp}\).
scientific article; zbMATH DE number 3873986